[Paper Review] Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

Cyril Touzé, Alessandra Vizzaccaro+1 more|arXiv (Cornell University)|Jul 11, 2021

Bladed Disk Vibration Dynamics355 references208 citations

TL;DR

This paper reviews nonlinear model order reduction (ROM) techniques for geometrically nonlinear structures, focusing on invariant manifold-based methods that use nonlinear mappings to capture complex couplings more efficiently than linear methods. The key contribution is a unified framework using parametrized invariant manifolds—particularly the center manifold and slow invariant manifold—that enable accurate, low-dimensional ROMs directly from finite element meshes without intrusive modifications.

ABSTRACT

This paper aims at reviewing nonlinear methods for model order reduction of structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes (NNMs) and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations (PDE). They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then the specific case of structures discretized with the finite element method is addressed. Implicit condensation, giving rise to a projection onto a stress manifold, and modal derivatives, used in the framework of the quadratic manifold, are first reviewed. Finally, recent developments allowing direct computation of reduced-order models (ROMs) relying on invariant manifolds theory are detailed. Applicative examples are shown and the extension of the methods to deal with further complications are reviewed. Finally, open problems and future directions are highlighted.

Motivation & Objective

• To address the limitations of linear reduced-order models in capturing strong nonlinear couplings in geometrically nonlinear structures.
• To provide a comprehensive review of nonlinear model order reduction techniques, especially those rooted in invariant manifold theory.
• To unify and clarify the theoretical foundations of nonlinear reduction methods, particularly the parametrization of invariant manifolds.
• To evaluate and compare existing techniques—such as modal derivatives, implicit condensation, and the quadratic manifold—within the invariant manifold framework.
• To promote the adoption of nonlinear, geometry-based ROMs for efficient simulation and design of complex nonlinear structures.

Proposed method

• Utilizes invariant manifolds as the geometric foundation for ROMs, ensuring that trajectories remain confined within the reduced space.
• Applies the parametrization method to compute center and slow invariant manifolds (SSM) via normal form theory and graph representation.
• Integrates finite element (FE) discretization with nonlinear reduction by deriving ROMs directly from FE meshes using non-intrusive approaches.
• Employs the quadratic manifold and modal derivatives as special cases of invariant manifold parametrization for computational efficiency.
• Uses the method of multiple scales and normal forms to systematically derive reduced dynamics on invariant manifolds.
• Demonstrates non-intrusive implementation via offline-online decomposition, enabling use with existing FE solvers without code modification.

Experimental results

Research questions

• RQ1 How can invariant manifold theory be systematically applied to derive accurate and efficient ROMs for geometrically nonlinear structures?
• RQ2 What is the relationship between established techniques like modal derivatives and implicit condensation and the more general framework of invariant manifolds?
• RQ3 In what ways do nonlinear mappings outperform linear basis enrichment in capturing nonlinear couplings in large-amplitude vibrations?
• RQ4 How can invariant manifold-based ROMs be computed directly from finite element models in a non-intrusive manner?
• RQ5 What are the computational and accuracy trade-offs between nonlinear manifold-based ROMs and classical linear or hybrid methods?

Key findings

• Nonlinear model order reduction based on invariant manifolds provides a geometrically consistent framework that ensures long-term dynamics remain within the reduced space.
• The parametrization of invariant manifolds unifies diverse techniques such as the quadratic manifold, modal derivatives, and implicit condensation under a single theoretical umbrella.
• Invariant manifold-based ROMs achieve high accuracy with fewer degrees of freedom than linear methods, especially in capturing non-resonant couplings.
• The method enables non-intrusive implementation with existing finite element codes, preserving versatility and computational efficiency.
• Recent advances allow direct computation of ROMs from FE meshes without time integration data, making the approach simulation-free and scalable.
• Despite higher computational cost in manifold computation, the resulting ROMs are more predictive and robust than linear alternatives, particularly in complex nonlinear regimes.